Are there infinitely many metalogics?

Question

Given the definitions of material implication, logical implication, and what a tautology is, we can prove: $$\mathcal B\text{ logically implies }\mathcal C\text{ if and only if }(\mathcal B\rightarrow\mathcal C)\text{ is a tautology.}$$ Clearly, this is also a proposition. However, if our meta domain also consists of true and false propositions, then we could also conceive of a meta meta logic that would prove the same theorem about the current meta logic, etc ... ad infinitum.

Am I correct in thinking this? Do we have no stopping point? If no, why isn't this problematic?

Given that the above statement is a proposition itself, why don't we dispose altogether of the concept of a metalogic and say that it is nothing more than a statement in the logic that we are formalising? At the end of the day, it is a $P\leftrightarrow Q$ type of proposition. Perhaps this could be a special consideration for our natural, psychological logic.

Answer 1

Rather than "expanding outward", I think it's more correct to think of this as "tunneling inward".

That is, there is just one meta level, which is the level of humans doing mathematics and writing arguments in natural language. In our mathematics, we can build a formal system called a logic which can reason about mathematical objects called propositions. This logic and these propositions are often constructed to model what goes on at the meta level. In this case, we can take meta level statements and "internalize" them to object level propositions in order to study them. But remember: statements at the meta level are human natural language constructs, while these object level propositions are abstract mathematical objects.

Now we are in a situation where at the meta level we can make statements about logic and propositions. So we can, if we like, internalize these statements to propositions, which we refer to "doubly internalized" propositions (let's call them 2-propositions). Now we can make statements at the meta-level about propositions about 2-propositions, which can again be internalized to propositions about 2-propositions about 3-propositions. Etc etc.

So we can, if we find it useful, tunnel to any depth we like of propositions talking about propositions talking about propositions. But at the end of the day we are here at the fixed meta level: humans reasoning about these complicated things in natural language human mathematics.

Answer 2

Broadly taken, a counterpart of this issue is the question of semantic ascent after W. V. Quine in philosophy. I believe it will be beneficial to expand on the issue in that perspective as well.

Let us set down the formal terminology (mostly because of shortcut usages, there are variations, even incoherences among authors):

• $A\rightarrow B$ is a proposition in the form of material implication. It is a well-formed formula of propositional calculus. The language of propositional calculus is the object language: Its objects are propositions, we express propositions and their relations through it.
• $\Gamma\vdash A\rightarrow B$ is a sentence in the metalanguage associated with propositional calculus. It states a judgement, that $A\rightarrow B$ is deducible from $\Gamma$. We express judgements about propositions and their relations through the metalanguage.
• $\Gamma\vDash A\rightarrow B$ is, likewise, a sentence in the metalanguage associated with propositional calculus. It states a judgement, that $A\rightarrow B$ is logical consequence of $\Gamma$.
• $\vDash A\rightarrow B\;\Rightarrow\;\vdash A\rightarrow B$ is also a sentence in the metalanguage associated with propositional calculus. It states a judgement, that if $A\rightarrow B$ is a tautology then it is a theorem of propositional calculus, which is a metatheorem.

Ascending to a higher level language (metalanguage) is an almost indispensable method; thereby we are able to state generalisations that otherwise (unless an alternative method is found) would not be possible. Examples are abundant, any general thesis about a system that contains infinitely many axioms (Peano Arithmetic, ZFC, . . .) cannot be stated by any finite sentence.

In Logical Syntax of Language, Rudolf Carnap makes a distinction between material and formal modes of speech. In the material mode, we talk mainly about the world, extra-linguistic reality; in the formal mode, we talk about the language that describes the world. Roughly speaking, Carnap's material mode of speech corresponds to the object language and formal mode to the metalanguage in the present context.

Thus, "$\omega$ is the first transfinite ordinal number" is a description in material mode; and "'$\omega$' is a name that belongs to the domain of numbers" is a formal specification.

Quine takes on Carnap's distinction and adds an ontological concern to it: Moving up the hierarchy of languages, we also move away from the talk about factuality, the objects and their relations we are empirically familiar to the talk that purports to refer to them in such a way as Bertrand Russell has said in Mysticism and Logic and Other Essays about mathematics: ". . . in which we never know what we are talking about, nor whether what we are saying is true."

Beside the usefulness of metalinguistic discourse mentioned above, Quine points out in Word and Object that semantic ascent is widespread device we come across under various appearances:

The strategy [of semantic ascent] is one of ascending to a common part of two fundamentally disparate conceptual schemes [compare to proof-theoretic and model-theoretic views of logical consequence], the better to discuss the disparate foundations. No wonder it helps in philosophy.

But it also figures in the natural sciences. Einstein's theory of relativity was accepted in consequence not just of reflections on time, light, headlong bodies, and the perturbations of Mercury, but of reflections also on the theory itself, as discourse, and its simplicity in comparison with alternative theories. Its departure from classical conceptions of absolute time and length is too radical to be efficiently debated at the level of object talk unaided by semantic ascent.

Formally, we can multiply the linguistic/theoretic meta-levels just as we can multiply the dimensions of a vector space. This poses a dilemma to us: While the ascent is a useful method, strategy, tool, it distances us from the realm where our primary concerns, in fact, reside. On the other hand, without it, we cannot get off the ground. Given the stage of the development of matters, a reconciliatory principle, if not its unfolding ways, can be that we may push on till the level up to which our generalisations keep meaningful and our abstractions keep truthful.

Answer 3

From a technical standpoint (we are not really talking about metamathematics, despite the tag) I don't agree with the answer from Alex Kruckman. There's isn't "just one meta level". And it isn't (necessarily) "the level of human doing mathematics and writing arguments in natural language".

Indeed - * discounting problems of cardinality and computability - we do have infinite "metalogics" for each "formal system". At each level we can name each object(type)/sentence/predicate of the previous level plus others of our choice, generate a "metalogic" and repeat.

This is not problematic - with the * caveat - just because we haven't found any consistency problem with this yet. Infact it's useful: we don't just use our "first level logic" because it has been shown that lots of predicates we would like to use (such as basic truth values, denotation relations, being provable, consistent, well formed, etc. just to name a few) can give problems if we don't separate a "world" from a "language". Or - in your terminology - a "logic" from a "metalogic".